A233206 Number of ways to write n = k + m (0 < k <= m) with k! + prime(m) prime.

0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 1, 5, 2, 3, 5, 3, 3, 4, 7, 4, 4, 6, 3, 3, 5, 6, 4, 5, 4, 4, 2, 4, 4, 7, 9, 4, 6, 5, 5, 5, 6, 8, 8, 7, 8, 6, 5, 5, 5, 7, 8, 7, 7, 8, 7, 9, 7, 6, 10, 6, 6, 9, 4, 7, 4, 9, 8, 8, 5, 9, 6, 2, 6, 7, 3, 8, 8, 9, 9, 7, 6, 10, 8, 8, 11, 7, 7, 4, 6, 8, 8, 5, 8, 5, 8, 14, 8, 7, 10, 8

Offset: 1

Author

Zhi-Wei Sun, Dec 05 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 3.

We have verified this for n up to 10^7. For n = 1356199, the least positive integer k with k! + prime(n-k) prime is 4496. For n = 7212995, the smallest positive integer k with k! + prime(n-k) prime is 4507.

Examples

			a(6) = 1 since 6 = 3 + 3 with 3! + prime(3) = 6 + 5 = 11 prime.
a(11) = 1 since 11 = 4 + 7 with 4! + prime(7) = 24 + 17 = 41 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000142, A231201, A231516, A231557, A231561, A231631, A233150, A233183, A233204.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[k!+Prime[n-k]],1,0],{k,1,n/2}]
Table[a[n],{n,1,100}]